적분법 전체 문제

Find the mass and the center of mass of $\rho =1$ on the region bounded by $y={x}^{5}$ and $y=\sqrt{x}.$

For the following exercises, find the requested arc lengths.

The length of $x$ for $y=\cosh (x)$ from $x=0\,\text{to}\,x=2.$

For the following exercises, find the requested arc lengths.

The length of $y$ for $x=3-\sqrt{y}$ from $y=0$ to $y=4$

The shape created by revolving the region between $y=4+x,$ $y=3-x,$ $x=0,$ and $x=2$ rotated around the y-axis.

The loudspeaker created by revolving $y=1/x$ from $x=1$ to $x=4$ around the x-axis.

For this exercise, consider the Karun-3 dam in Iran. Its shape can be approximated as an inverted isosceles triangle spanning across the river, with height 205 m and width (across the top of the dam) 388 m. Assume the current depth of the water is 180 m. The density of water is 1000 kg/m³. Find the total force on the wall of the dam.

You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and $45^\circ \text{F}$ outside and the temperature of the body is $78^\circ \text{F}.$ You know the cooling constant is $k=0.00824^\circ \text{F/min}.$ When did the victim die, assuming that a human’s temperature is $98^\circ \text{F}$ ?

[T] The best-fit exponential curve to these data is given by $y=40.71+{1.224}^{x}.$ Why do you think the gains of the market were unsustainable? Use first and second derivatives to help justify your answer. What would this model predict the Dow Jones industrial average to be in $2014$ ?

Find the volume of the catenoid $y=\cosh (x)$ from $x=-1\,\text{to}\,x=1$ that is created by rotating this curve around the $x\text{-axis},$ as shown here.

Find surface area of the catenoid $y=\cosh (x)$ from $x=-1$ to $x=1$ that is created by rotating this curve around the $x\,\text{-axis.}$

In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals.

$\int {x}^{3}{e}^{2x}dx$

In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals.

$\int {x}^{3}\ln (x)dx$

In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals.

$\int {y}^{3}\cos ydy$

In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals.

$\int {x}^{2}\arctan x\,dx$

Find the integral by using the simplest method. Not all problems require integration by parts.

$\int v\sin vdv$

Find the integral by using the simplest method. Not all problems require integration by parts.

$\int \ln x\,dx$ (Hint: $\int \ln x\,dx$ is equivalent to $\int 1\cdot \ln (x)dx.)$

Find the integral by using the simplest method. Not all problems require integration by parts.

$\int x\cos x\,dx$

Find the integral by using the simplest method. Not all problems require integration by parts.

$\int {\tan }^{-1}x\,dx$

Find the integral by using the simplest method. Not all problems require integration by parts.

$\int {x}^{2}{e}^{x}dx$

Find the integral by using the simplest method. Not all problems require integration by parts.

$\int x\sin (2x)dx$